mathematical induction qu

Hey guys theres this inequalities mathematical induction question which I absolutely cannot do. Could someone please give it a go cos there arent any answers in the book. Ive tried everything I can think of, even expanding (k+1)^5 but I still cant get it, even tho I can see that its true just from common sense.

Re: mathematical induction qu

Re: mathematical induction qu

The question:-

Prove that 5^n ≥ n^5 for n ≥ 5.

Proof:-Let n=5.LHS=5^6=15625RHS=6^5=7776, hence LHS>RHS.

Let n=7.LHS=5^7=78125RHS=16807, hence, LHS>RHS.

Let 5^k >k^5 where k≥ 5.Lets compare 5^(k+1) and (k+1)^55^(k+1)=5 x 5^k(k+1)^5=k^5+5K^4+10K^3+10k^2+5K+1=5(k^5/5+k^4+2k^3+2k^2+K+1/5)Cacelling 5 on both sides, the LHS is 5^k andthe RHS is k^5/5+k^4+2k^3+2k^2+K+1/55^k > k^5/5+k^4+2k^3+2k^2+K+1/5Since the inequation is true for k+1 when it is true for k,it is said to be true for any k≥5.